Living, as I do, with a football mad son, its difficult to avoid the men's football World Cup. I have kept up to date with the scores and watched the odd highlights, but for the first time, the other evening, sat down to watch a full 90 minutes. My mind started to wander; how many different final tables can there be in the group stages?
4 teams, 3 games each, 3 points for a win, 1 for a draw. What might the answer be? Lots of 4 and 3s, possibly 12 different final tables? 4x3x2x1 = 24 maybe?
What strategy should be employed? Let’s start with the first team winning all three games. The second team winning their other two games. The third team winning their last game. The fourth team not winning any. Let’s call this a (9,6,3,0) table. Other options starting with 9: (9,6,1,1) (9,4,4,0) and so I went on.
I now looked at tables starting with the winning team winning two games and drawing the third producing final tables such as (7,7,3,0); (7,7,1,1)…
I soon passed my prediction of 12 different tables. Tables starting with 6, then tables starting with 5 took me well past the 24 prediction. Then a couple of unusual tables – all four teams winning one game, drawing one game, and losing one game: (4,4,4,4) – all games being drawn (3,3,3,3). I am now well beyond 30 different possible tables.
How do I check I have got them all? Have I included any tables that are not possible? I looked back through group tables from previous World Cups, ticking off those I have got, adding others combinations I have missed. Some final tables appear quite frequently, others I am yet to find: has (5,5,5,0) ever happened?
Pictured at the top of this blog is some of my working out. Have I got all the possible combinations? Have I got any combinations that are impossible? There is your homework for next week! The game I was watching? Not sure what the final score was – or who was playing!
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